FINITE DIFFERENCE METHODS FOR NEARLY SINGULAR PROBLEMS
Abstract
Partial differential equations are an important tool in the analysis of the behavior of physical systems. This thesis presents a new approach to the solution of such equations based on the HODIE method of Lynch and Rice for elliptic boundary value problems. The new method is designed to be most effective on problems which are termed "nearly singular." Several applications involving problems of this type are given as motivation for this research, followed by a survey of existing methods for both singular and nearly singular problems. Because their lower dimensionality makes them easier to handle, and because they are often approached in a manner computationally similar to elliptic equations, two-point boundary value problems in ordinary differential equations are first considered. A theoretical justification of the new technique notes that it reduces the constant, not increases the exponent, on the leading term of the truncation error. The usefulness of the method depends on the efficient generation of a basis function which models the nearly singular behavior of the solution; several schemes for accomplishing this are discussed. The HODIE method using one or more special basis functions is then presented, along with a discussion of the overhead they require. The effectiveness of the technique is demonstrated and its sensitivity to disturbances in its parameters is investigated in a series of numerical experiments involving both one and two-dimensional Dirichlet problems.
Degree
Ph.D.
Subject Area
Computer science
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